This means that, if k.k is a norm on X, such that (X,k. Completeness for a normed vector space is a purely topological property. As \(U_1\) is open, \(B(z,\delta) \subset U_1\) for a small enough \(\delta > 0\). A Banach space over K is a normed K-vector space (X,k.k), which is complete with respect to the metric d(x,y) kxyk, x,y X. We can assume that \(x x\), then for any \(\delta > 0\) the ball \(B(z,\delta) = (z-\delta,z+\delta)\) contains points that are not in \(U_2\), and so \(z \notin U_2\) as \(U_2\) is open. Suppose that there is \(x \in U_1 \cap S\) and \(y \in U_2 \cap S\). We will show that \(U_1 \cap S\) and \(U_2 \cap S\) contain a common point, so they are not disjoint, and hence \(S\) must be connected. a countable product of unbounded metric spaces with a metric that acknowledges the boundedness characteristics of the factors. 1 Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The distance is measured by a function called a metric or distance function. The word 'bounded' makes no sense in a general topological space without a corresponding metric. In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. Conversely, a set which is not bounded is called unbounded. \), \(U_1 \cap S\) and \(U_2 \cap S\) are nonempty, and \(S = \bigl( U_1 \cap S \bigr) \cup \bigl( U_2 \cap S \bigr)\). In this paper we fix 1\le p<\infty and consider (\Om,d,) be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure supporting a. In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure.
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